Skeleton-based zero-shot action recognition aims to recognize unknown human actions based on the learned priors of the known skeleton-based actions and a semantic descriptor space shared by both known and unknown categories. However, previous works focus on establishing the bridges between the known skeleton representation space and semantic descriptions space at the coarse-grained level for recognizing unknown action categories, ignoring the fine-grained alignment of these two spaces, resulting in suboptimal performance in distinguishing high-similarity action categories. To address these challenges, we propose a novel method via Side information and dual-prompts learning for skeleton-based zero-shot action recognition (STAR) at the fine-grained level. Specifically, 1) we decompose the skeleton into several parts based on its topology structure and introduce the side information concerning multi-part descriptions of human body movements for alignment between the skeleton and the semantic space at the fine-grained level; 2) we design the visual-attribute and semantic-part prompts to improve the intra-class compactness within the skeleton space and inter-class separability within the semantic space, respectively, to distinguish the high-similarity actions. Extensive experiments show that our method achieves state-of-the-art performance in ZSL and GZSL settings on NTU RGB+D, NTU RGB+D 120, and PKU-MMD datasets.